Adjustable Piecewise Quartic Hermite Spline Curve with Parameters

In this paper, the quartic Hermite parametric interpolating spline curves are formed with the quartic Hermite basis functions with parameters, the parameter selections of the spline curves are investigated, and the criteria for the curve with the shortest arc length and the smoothest curve are given. When the interpolation conditions are set, the proposed spline curves not only achieve C1-continuity but also can realize shape control by choosing suitable parameters, which addressed the weakness of the classical cubic Hermite interpolating spline curves.

C˜2 Continuous Cubic Hermite Interpolation Splines with Second-Order Elliptic Variation

In order to solve the deficiency of Hermite interpolation spline with second-order elliptic variation in shape control and continuity, c-2 continuous cubic Hermite interpolation spline with second-order elliptic variation was designed. A set of cubic Hermite basis functions with two parameters was constructed. According to this set of basis functions, the three-order Hermite interpolation spline curves were defined in segments 02, and the parameter selection scheme was discussed. The corresponding cubic Hermite interpolation spline function was studied, and the method to determine the residual term and the best interpolation function was given. The results of an example show that when the interpolation conditions remain unchanged, the cubic Hermite interpolation spline curves not only reach 02 continuity, but also can use the parameters to control the shape of the curves locally or globally. By determining the best values of the parameters, the cubic Hermite interpolation spline function can get a better interpolation effect, and the smoothness of the interpolation spline curve is the best.

Irregular Symmetrical Object Designed By Using Lambda Miu B-Spline Degree Four

In Computer Aided Geometry Design (CAGD), B-splines curves are piecewise polynomial parametric curves that play an important role. CAGD involves the interpolation and approximation curves and surfaces. CAGD has been widely used which brings good impact of computers to industries in manufacturing. There are many improved methods in the B-spline curve such as extended cubic B-spline, trigonometric B-spline, quasi trigonometric B-spline, and λμ-B-spline. Each of the methods has its behaviour and advantage. In this paper, λμ-B-spline was used to be implemented in generating irregular symmetrical objects. λμ-B-spline has a shape parameter that can change the global shape by manipulating the value of the shape parameter. The bottle has been chosen as an irregular symmetrical object. The 2-dimensional symmetrical curves of Bottle design were formed by using λμ-B-spline degree 4. The curves designed are dependent on the shape parameter which can be adjusted. Then, the curves generated were revolved using the Sweep Surface method to form 3-dimensional objects. Every object has its volume and this research focused on the numerical method which was Simpson’s 3/8 to compute the volume. The volumes obtained were compared to the actual volume to determine the best shape parameter used. The results show that the λμ-B-spline curve with a shape parameter of 1 is the best shape parameter in designing symmetrical irregular objects with the desired volume.

Geodesic Hermite Spline Curve on Triangular Meshes

Curves on a polygonal mesh are quite useful for geometric modeling and processing such as mesh-cutting and segmentation. In this paper, an effective method for constructing C1 piecewise cubic curves on a triangular mesh M while interpolating the given mesh points is presented. The conventional Hermite interpolation method is extended such that the generated curve lies on M. For this, a geodesic vector is defined as a straightest geodesic with symmetric property on edge intersections and mesh vertices, and the related geodesic operations between points and vectors on M are defined. By combining cubic Hermite interpolation and newly devised geodesic operations, a geodesic Hermite spline curve is constructed on a triangular mesh. The method follows the basic steps of the conventional Hermite interpolation process, except that the operations between the points and vectors are replaced with the geodesic. The effectiveness of the method is demonstrated by designing several sophisticated curves on triangular meshes and applying them to various applications, such as mesh-cutting, segmentation, and simulation.

Two Extensions of the Quadratic Nonuniform B-Spline Curve with Local Shape Parameter Series

Two extensions of the quadratic nonuniform B-spline curve with local shape parameter series, called the W3D3C1P2 spline curve and the W3D4C2P1 spline curve, are introduced in the paper. The new extensions not only inherit most excellent properties of the quadratic nonuniform B-spline curve but also can move locally toward or against the fixed control polygon by varying the shape parameter series. They are C1 and C2 continuous separately. Furthermore, the W3D3C1P2 spline curve includes the quadratic nonuniform B-spline curve as a special case. Two applications, the interpolation of the position and the corresponding tangent direction and the interpolation of a line segment, are discussed without solving a system of linear functions. Several numerical examples indicated that the new extensions are valid and can easily be applied.

Optimal Intersection Curves for Surfaces

In this article, an algorithm has been established to approximate parametric-parametric, explicit-implicit, and explicit-explicit surface intersection. Foremost, it extracts the characteristic points (boundary and turning points) from the sequence of intersection points and fits an optimal cubic spline curve to these points. Moreover, this paper utilizes genetic algorithm (GA) for optimization of shape parameters in the portrayal of cubic spline so that the error is minimal. The proposed algorithm is demonstrated with different types of surfaces to analyze its robustness and proficiency. In the end, all illustrations show the effectiveness of the algorithm which makes it more influential to resolve all complexities arises during intersection with a minimal error.